Physics 105 Problem Set 9
Due: Thursday, November 29, 2006, by 4:30 pm in 208 Jadwin.
Required reading for this week:
-- Knight Chapters 20 and 21
We understand that we blasted through waves rather quickly.
Though we discussed the fundamentals in lecture, we want you to concentrate
on the "phenomenology". From your reading and our work in precept,
try to get used to the properties of travelling and
standing waves, the parameters of sinusoidal waves, and how superposition
-- It is a very good idea to
warm up by doing problems in 103 Learning Guide 10 after you've done the
required reading. Problem IX tells you why we spend so much time with
sines and cosines.
Reading for NEXT week:
We will be spending the rest of the term on thermodynamics. This is not
covered in K&K, so we will be reverting to Knight and
Fermi's "Thermodynamics." In the last 2½ weeks, we will cover
-- Knight Chapters 16-19 (you have seen this before), Fermi
Sections 1-14 (this will be new)
Additional problems to turn in.
Problem 9.1 A wave is represented by
Z1(x,t) = 10 cos(k x + ω t)
Show that this represents a travelling wave and deduce its wavelength,
frequency, speed and direction of travel.
A second wave
Z2(x,t) = 10
cos(k x + ω t + π/3)
interferes with the first wave. Deduce the amplitude and phase of the
Problem 9.2 You are building an electric guitar and
have cut 24 frets (places to clamp your finger down on a string, effectively
changing its length) out of fret wire; now you are wondering where
they should go on the neck of the guitar. The distance from nut to bridge
(where the ends of the guitar strings are fixed)
is L = 65 cm.
- a) How far away
from the bridge should you put the 12th and 24th fret? (The 12th and 24th
frets are where you play the first and second octave i.e., double and
quadruple the frequency of the open string.) Optional:
Find a formula for the positions of all 24 frets.
- b) The note
a (440 Hz) can be played by pressing the g-string to the
neck at the second fret (57.9 cm away from the bridge) and plucking the
string. What is the wave speed on the g-string?
- c) What is
the frequency of the harmonic that you can play by plucking the
g-string while slightly touching it with your finger
approximately above the 7th fret (2/3 L away from
the bridge) and thereby
putting a node there?